Covariant quantization of totally antisymmetric tensor-spinor field in $AdS_d$

We develop the quantization of a recently proposed model describing a totally antisymmetric rank-$p$ tensor-spinor field (a fermionic $p$-form theory) in $d$-dimensional anti-de Sitter (AdS) space. The model provides a new nontrivial example of a reducible gauge theory, in which gauge transformations are linearly dependent and the degree of reducibility increases with $p$. It is well known that in such cases the standard Faddeev-Popov-DeWitt prescription for the generating functional is not applicable. We quantize the fermionic $p$-form theory using the general Batalin-Vilkovisky (BV) formalism, employing two distinct gauge fermions associated with gauge-fixing functions of different admissible ranks, confirming the independence from the gauge choice. As a result, we obtain the quantum effective action in terms of a sequence of functional determinants corresponding to specific Dirac-like operators on AdS space.